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NUCLEAR RELAXATION STUDY OF MOLECULAR 
MOTION IN LIQUID AND SOLID AMMONIA 



ACKNOMI.EDOJESIS 



Co write: Difficult hocai 



of e dissertation, this is Che most difficult 






committee. Dr. t. A. Scott, for suggesting this study and providing a 
numerous very helpful suggestions throughout the course of this work. 






interesting and informative conversations, and to Mr. Basil McDowell 
Among the people who had no influence on what is written in this 









Setzer Cardan, for encouragement through many 



trying periods, and for careful proof-reading of the manuscript. 

Finally, I wish to thank Mrs. M. Beth Sercombe for her diligent 
proof-reading and highly competent typing of the final manuscript. 




s 



LIST OF TABLES 







(a.) temporal development of M *(c) following a 

90" pulse ' 






pc ra cur el 










LN <R(t^ 8 va. time/T' 



Theocccical proton T® reorientation correlation . 



Logj^jCTp vs. lO-Vitenperatnre) . 





















35. typical parallel tuned sample coil circuit 



CHAPTER I 
INTRODUCTION 


















Nuclear Resonance techniques nay be 



gorles: Transient and continuous wove NMR. Within the framework of 
form of a lineshape, or frequency distribution, function; whereas in 




is particularly critical in a lineshape e 
to be discussed in some detail in Chap. 1 






models of molecular motion nicely, ore available from the scientific 




interesting electronics problem. 



indicated by experiment, and Pauling cd 







Fig. I. FREE 



AMMONIA MOLECULE 



i considered a discorcion of Che 2s orbital forming 
intermolecular bonds of an ionic nature (2) known as hydrogen bonds. 

kcal/mole N-H bond energy (4) gives one some perspective to Che magni- 









The objective of Chi 
iasons, but primarily be 



is is not a trivial task for a multitude ol 



sarily involves an exchange of energy, thereby altoring the normal 

able compromise with Nature: We compromise the ability to study a 
single molecule and look instead at the average motion of an effective- 






a magnetic field it, the nucleus 



E - -ytan | Sj (2.2) 



» - t-I, -CI-1), — •. (2.3) 

end y is a constant of proportionality, called Che gyromagnedc ratio. 
















he molecular motion and the results obtained by using these 






characteristic of 






The Hamiltonian O: 
and the individual te: 






is examined in some detail. The general Hamil- 
rstem la written out explicitly in Sllchcer (6). 



Hamiltonian that we work with will provide a back door through which 
are now written out. Since ultimately we ere interested in describing 












given in Eq. (2.2). Rocaclon of Che molecule docs noc affect this 
t is spatially and temporally constant. 



2.3.2. Buclear Dloole-Dlnole Interaction Tern 




Eq. (2.5) 



o -‘ > 

V - ” 1 * 

•--K^ 4 V5“'V”“*'“ lk 

—f'S-V*’" 



««■»*- 






. ...» .. H, ... H . ,.i„- 






E,. (2.7) 



Tacts fron the external field 9 . Note the 



Andrew (8). 









was first considered in Che classic Bloembergen, 



magnetic Held ae a nuclear site In a rigid lattice. Van Vleck (10) 
has investigated this problem in detail and derived an expression for 
calculating the mean square local magnetic field at a given nuclear 

nuclei in the solid are allowed. The effect of rapid, isotropic mo- 
lecular motion on terms A and B is to average the angular factor to 



aero. Theories have been developed 
field from the rigid lattice value 



Co molecular motion. 



To this point we have considered oi 
moment along an externally applied magnetic fie 



electric field gradi- 



excepclon: In diamagnetic materials cl 
term whereas Che molecular electric fie 



the sample without breaking 
ilar field gradient at the 




K o - fusTTi < 31 ! - ** * * *■*». 










V 






A point of interest in this Hamiltonian is that the I £ operator appears 
os a squared tern leading to degenerate eigenenergies for the m - +1 
states. The factor eq in Eq. (2.13) is dependent on the environment of 

property of the nucleus itself. It is conventional to define Che term 
(eqQ/h) as the quadrupole coupling constant expressed in Ha. The 
quadrupole coupling constant for the case of nitrogen in ammonia is 

-3.16 KHz at 77 K to -3.08 MHz at 193 K (approximately 3 K below Che 
normal melting point). The quadrupole Hamiltonian is thus approxi- 

treated easily as a perturbation as can 

2.3.4. Magnetic Interactions of Hudel with Molecular Electrons 
The Zeeman Hamiltonian Introduced in Sec. 2.3.1. is rigorously 

consider the magnetic Interactions possible in a diamagnetic molecule 

actions of this nature are separable into two broad categories: 

externally applied magnetic field, and b) Those 

externally applied magnetic field, lt 0 , which has the 



site. The contribution of R + fl . along fi nay be function of the 
relatlvo orientation of the molecule with H . This effect is usually 

of the component of 6. + R along fl affects the Zeeman Hamllconian. 



each molecular species is exploited tc 
in both quantitative and qualitative analysis of 









of a few hundred parts per million would obviously totally wash out 



liquids the dipolar contribution to the nagnetic field at a nuclear 




Interactions of type b) prove to be quite interesting in the study 

Maxwell (16) and Gucowsky and McCall (17). The paper by Hahn and Max- 
well is particularly interesting because transient NMR techniques ere 
used. It was found experinentally chat this interaction was lodcpend- 

K ln . hJlj-Ij. (2.16) 





Experimentally, we are able to consider the various Interactions 






tally separate experiments because the wide difference in magnitude 

perlmental apparatus to be sensitive only to small non-overlapping 
frequency intervals at the ccaccr frequency corresponding to a Zeeman 


















^ with the nitrogen nucleus and b) information obtainable by 



) Indiroct spin-spin interaction 
molecular proton. 















Intramolecular dipole-dipole interaction among 






CHASTER III 

INTRODUCTION TO NUCLEAR MAGNETIC RESONANCE 
AND RELAXATION 










*0 - H ■ 3fcpV«X#l)« (3. 

The magnitude of Che scacic nuclear magnetic susceptibility, x 
Is typically lo" 10 to 10' 11 . Therefore, for a laboratory field of 

trivial experimental problem using static techniques. Static ccchr 
ques are difficult experimentally and not very fruitful when one la 
interested in the study of molecular motion, thus other methods mua 



3.2. The Nuclear Magnetic Besonance Phenomenon 
»e have already established that the effect to be measured is qui 
small and Is likely to require correspondingly sophisticated experi- 
mental techniques. Highly elegant techniques have been devised to 

nal which Is amplified in some cases over 
as any signal which has no correlation wi 



in electrical slg- 






snall electrical sig- 
iefine quite generally 
nuclear signal. This 
experimental techniques; so for 



lar momentum as well as a magnetic moment. The equilibrium nuclear 
moment but also a total static angular momentum X which Is related 



These quantities are of 



id quantum mechanically. Although 



what follows will appear strictly classical, i 
given that l! - II £ and at t ■ 



. The temporal development 



" we transform Co a coordinate system with a cod 
ling the relationship on p. 133 of Goldstein (2! 






a - s - - t B 5 






ation except we enclose the sample in a coil of wire, stationary in the 
lab frame with its axis normal to It , then a voltage is induced in the 

coil as a sinusoidally varying change in magnetic flux with a funda- 

ducer, mentioned earlier, from which we obtain the signal voltage. This 
is of course a nonphysical situation os the Boltzmsnn distribution is 

ately begins to seek equilibrium. Furthermore, we have assumed that 

of 8 small, laboratory generated perturbation to the Zeeman Hamiltonian, 
let the perturbation be the application of a voltage. 



[3. 9) 



to the coil surrounding the sample. For geometric simpllficetion we 
position the coil as shown in Fig. 3, with its axis along the y axis 

fij - -JlHjSinwt, <3.10 






component rotating in a sense opposite Co chat of the free precession 
of the nuclear spin system has negligible effect on the spin system 



rotating in such a manner that it lies out the x axis. Our initial 
conditions are such that at t » 0, H ■ M 1c and both Che lab and ro- 



(^j* - B*xy[B 0 + “)1!* + Hifl. (3.11) 










2yH,' 



Generalizing, Che pulse duration fc 






He have implicitly assumed chac is approximately uniform over 

possess certain lower bounds which depend on Che sample studied. 

complete. To measure the equilibrium magnetization of Che spin system: 
He simply allow the spin system Co come to equilibrium immersed in an 
applied magnetic field, apply a z/2 pulse to the coil thereby rotating 
down into the x,y plane, then monitor the voltage induced in the 

le electronics, |H | may be calculated. 






:. Transient N> 









wave NMR (CtfNMR) and transient 1EO (TNMR), as Che response 
ence in the two methods arises primarily from differences i 



sal equilibrium is noc appreciably affected. The absorption of energy 

applies rotating ^ fields of such a magnitude that the Boltsmann pop- 
the following two distinct experimental steps: 




distribution function centered about H with g(H)dH defined as the 



/g(H)dH - 1 . 



(3.18) 




8* - eah* + a/. (3.i9) 

s width of the distribution function g(H). W 

about Hjd down to the -‘■>’1 axis. This condition is sometimes not 

large dipole coupling sometimes present, but in liquids it is usually 
quite easy to satisfy. For this discussion we assume this condition 



any significance ft 






Fig. 5(a.)dcpicts Che situation os seen j 



• 90° pulse. Fig. 5(b) 





Fig. 5. 'IDEAL' SPIN 



SYSTEM BEHAVIOR FOLLOWING A 90 - PULSE 




ble migration co Chennai equilibriun. We noce Chat scadscical chcr- 

pulse has Che e£fecc of equalising Che spin scace populacions and cre- 

pulse Che veccor sun of Che nuclear nagnecizacion/unic volume is 
ft(o) * • The cwo processes leading Co ulcfmaee descruccion of 

processes is implicitly concained in Che lineshape function, f(u). 










M y *(T) - » [ 






scribing loss of crsnsverse nagnetleotion. Note that implicitly 
this Includes processes governed by T^. Thus the lineshape function 







DETECTED SAMPLE 






G(t) ■ / f (v)cos 2svt 2 i 



lv ■ 2r/f(v)EXP(-2niv)dv. 



multiply through hy EXP(lui't). 

EXFClu't) /f (ui)EXP(-iut)du “ ^V, (t)BCP(liu't) (3.31) 

/duf(u) /EXP[-lt(im')ldt - - /V (t)EXP(iw't)de. (3.32) 

/EXP(-it(u-ti'}]dt - 2ir6(W), (3.33) 

J(m')-jJj /V d (t)EXP(lu't)dt . (3.34) 

Thus wo have demonstrated that the FID and the distribution function 






previously defined. 



f(cd) is defined by 



M n " (3.35) 

Mote that all odd moments vanish as a result of the even parity of 
f(u). The importance of determining the moments of a lineshape is 
twofold : a) Knowledge of the various lineshape moments provides de- 
tailed shape information; b) Van Vleck's (10) classic calculation re- 
veals the equality of Che mean square local field at a nuclear site 






ly calculate the second mo- 



actioos with neighboring I and S spins. This equation is valid only 
if all spins are in equivalent lattice positions. 



2 3Tjfi 2 I(I+l) ^ (1 - 3cos 2 6 1k ) 2 

I species 




whore r^ fc are the respective intranuclear distances. This calculation 
is quite tedious for nuclear configurations of little symmetry. Great 
simplifications in this problem are usually possible however thanks Co 
the rapid r decay of the squared dipole-dipole interaction. One may 

ocher simplification occurs for the case of a powdered sample, for 
which Eq. (3.36) averaged over all angles bccoces, 






•&¥»«> i •?. 

™,SI. 




-■ 




l(v * V 



- JltotCv - xHx, 



..-te 






ir polycrystalline sample i 



my cases, satisfactorily represented by a Gaussian 
le normalised Gaussian curve is given by 



le associated FID by substitution o 



Eq. (3.43) into i 



(3.45) 



motion is very rapid, the Lorentaian lineshape is sometimes a better 
approximation to the experimental curve, Eq. (3.29) gives the follow- 
ing relationship for the FID when the Lorenczian lineshape function 



: defined in Eq. (3. 26) at 

'(b) illustrates f(u) obtained by the method of CUE 
empirically fit as well as possible to 





FREQUENCY (kHz) 



Fifl- 7(b), SOLID NHj 



LINESHAPE FUNCTION 



THREE TEMPERATURES 







, - .-ti 









for the coaplete spin systea to lose 
cessional state for a given nucleus is T_, as previously defined. Each 

sites , i.e. , we expect different values of w rna for different types of 







which contribute to the averaging process and further narrow the lino. 






!r angular factors appearing 


















ability of high energy Zeenar 
The Lorentaian shape function is 















hi; 








k -i?-«hv 



4 ' 





AMMONIA (PROTONS) 



liquids , 









lecular self-diffusion through an inhomogeneous magnetic field. 

















»(+)»(+-) - X(-)«(-f), 




- -2n[N(+)W(+-) - »(-)W(-«). 
Using Eg. (3.59), we find 



^--2W„[N(+) - N(-)e'*V kT ]. 

Eg. (3.59) has the value, 

VW» 3,2 , io- 4 












and (3.64) we see that the populations of the two 
thermal equilibrium are very nearly equal. Doing the 



id that Bq. (3.65) reduces tt 






approximations, nevertheless provides an excellent working approxima- 
tion for determining Tj, even in solids. The most fundamental assump- 
tion used iu the derivation of Bq. (3.69) is that which permitted the 
use of statistical techniques, l.e. that between any two spins there 

interaction, and on the average is temporally indistinguishable from 
any other possible spin-spin interaction. This assumption is rigor- 
ously satisfied for the magnetic dipale-dipole interaction between 









because the motion of these spies is clearly correlated. & great deal 
decay constants and different weighting factors. Eq. (3.69) is, how- 



INTERPRETATION OP EXPERIMENTAL RESULTS 
4.1. Outline of Experimental Results 

mention of tile method used. A thorough description o 1 






If Linevidth of CWNMR Absorption Curve 
X. Region of temperatures studied - -1 to 195 K 

3. Method of measurement - CWNMR using magnetic 

Calculation of Second Moment of CWNMR Lineshapes 

1. Region of temperatures studied - —1 to 195 K 

2. Magnetic field - 1.7 kG 



- Calculated from CWNMR llneshapes 



Measurement of vs. Temperature for Zeeman 



1. Region of Temperatures studied - 195 Co 239 K 






1. Region of Temperatures studied - 195 to 239 X 

2. Magnetic field 



. Measurement of T 2 vs. Temperature for Protons 
1. Region of Temperatures studied - 195 to 230 K 




All work reported herein was performed at an almost constant pres- 
sure over Che sample of —20 lb/in 2 absolute. The normal melting and 
boiling points of NH^ (1) are respectively 195.36 and 239.76 X. The 
anomalously high boiling point for a member of the Group V hydrides 
is attributed to the strong hydrogen bonding characteristics of the 






The solid phase crystal structure of Nil. 
rained using single crystal and powder X-ray 

be considered to be a slightly distorted fee lattice with each molecule 












neighbors — all molecules are in equivalent positions. The distance 










boor, reported In Che literature (31,32). The mccastable ferns occurred 
when NH 3 was deposited from Che gas phase onto a surface which was kept 
at approximately 77K. Our experiment was performed in such a manner as 
to exclude any possibility of forming a metastable phase. 

A complimentary lattice structure study of polycryatalline M> 3 was 
made by Seed and Harris (R-H) (33) using neutron diffraction techniques. 
Neutron diffraction studies give average positions of the nuclei whereas 
X-ray diffraction reveals maxima in the electron cloud distribution. He 
expect the neutron diffraction measurements Co be of more use because 
it is the average internuclear distance chat is associated with NMR 

N-N distance are within experimental 



1.005 * 0.023 8 is much 
* 0.004 8 (1,34) than ti 






X-ray powder work has been done on NHg to 4.2X (32) and no crys- 
l structure change from the high temperature structure is observed, 
it capacity measurements from 15K Co the vaporization point (35) 
'eal a sharp transition only at Che melting point. The heat oa- 
iity data do display subtle inflections . in the curve at approxi- 

et of different modes of molecular motion, a presumption supported 
this work. Thormal conductivity of solid NH 3 in the temperature 



. but the thermal expansion data of Menzhelll and Tolkect 
temperature region 24 to 175 K are Interpreted taking ir 






molecules. 



(37), a. 









work. Hydrogen bonding of 






kilocalories per mole. Torsional vibrational moi 
to molecular torsional oscillations about axes n< 

A quite interesting and important experiment performed by Lehrer 

as a function of temperature from 77 K to the malting point and the 



NH 3 and SDj. It is pointed out that torsional vl 
molecular symmetry axis, which is also the axis of the field gradient 
tensor, are not effective in reducing the quadrupole coupling constant; 



theory in conjunction with Reding and 



Hornig's (38) va] 

between molecular reorientation and torsio 
amplitude libratlons of a molecule within 
well are referred tf 



frequency given an excellent f] 
interpret these results to mean that lit 



I clarify the distinction 
s crystalline potential 



is torsional oscillations. Molecular reorienta- 
■°n will refer to a physical rotation of the molecule, e.g. a rota- 
■on of the NHg molecule about its C 3 axis through 2*/3 radians — from 
is position of stable equilibrium to another. 

The structure of liquid has been determined by X-ray diffrac- 
on work performed by Kruh and Pets (39). The liquid was studied 



.1 temperatures. They interpret tt 



radial distribution function at 199 K to indlcste t[ 
an ammonia molecule has eleven neighbors, seven at £ 

3.56 X and four at 4.1 X. The liquid structure beat 

the interesting speculation of a highly ordered liquid state when 
confronted with these results, a speculation not supported by nuclear 
relaxation results. Seoricntation in the liquid state is indeed quite 


















The application of detailed theory to the experimental 
immonia io order to aeparate Che varioua contributions 

cusston applies both to the solid and Che liquid state. The physical 
difference between the molecular liquid and solid scaces is character- 
ised by the inodes of molecular motion which are thermally activated. 

account of the problem chan chat given here may be found in lectures 
by Powles (40) and Bloom (41,42). 

The problem at hand is quite formidable: To separate the various 
contributions to tha spin lattice relaxation time, T^ P , and interpret 
what mode of molecular motion leads to a specific contribution. The 
separation is conveniently effected through the assumption that 
Eq. (3.69) may be written in the form: 

„ , , / M - M(t-O) , 



Proof of the validity of this assumption is not j 
retically, but empirical confirmation leads to v( 
Che assumption is widely used. It is obviously t 









A. ) l/xj ntra d - the intramolecular dipole-dipole term: This tern 

ecule. It seems quite generally valid to assume that the molecular 
bonds are rigid; therefore, magnetic fluctuations which stimulate Zee- 
man transitions are caused strictly by fluctuations in the angle 8 

to intermolecular dipole-dipole interactions, but in liquids both terms 
are typically of the same order of magnitude. 

B. ) l/l!°" C d - che intermolecular dipole-dipole interaction 
term: The magnetic field fluctuations in this case are produced by 
dipole-dipole interactions <Eq. «.6» between spins on different mole- 
cules. If molecular self-diffusion and reorientation are completely 
independent, the effect of this term is governed only by che frequency 
and magnitude of molecular collisions. This term will Chen contribute 
information concerning molecular self-diffusion. 

C. ) 1/T^ - The spin rotacion interaction term: The spin rotation 

che magnetic field produced by a rotating molecule. Fluctuations in 
this magnetic field caused by collision modulated molecular rotational 
states, i.e., changes in che molecular J quantum nueber, stimulate 
nuclear Zeeman transitions if the fluctuations ore rich in the tarmor 
frequency Fourier component. Ibis term is usually important only 
around the critical point of che liquid. For the case of ammonia 
1/T^ r has been measured by Smith and Powles (43) and found to be of 
little importance in the normal liquid region. Thus we are justified 
in not considering the spin-rotation interaction in this work. 

D. ) 1/Ti - The electric quadrupole interaction term: If a nuclear 



occur about Che symmetry axis of the molecule. This presumption is 
well supported by the l \ <iuadrupole resonance work of lehrer and 
0 Konski (14). We thus consider the following model: The proton 
nuclear relaxation is governed exclusively by inter- and intramolecular 
dipole-dipole interactions modulated by the molecule undergoing hin- 
dered rotations about the symmetry axis to the three possible posi- 
tions of stable equilibrium. 1 
be a stationary random process 
don time t c , which is roughly 



:h may be characterized by a correla- 






r c obeys an equation ol 



e form (see Abragara p. 






: che barrier hlnder- 



stant, and E fl is Che activation energy 
calories/molo. Eq. (4.4) is called the 

As a first approximation to Eq. (4.3) we assume all terms on the 
right hand side are negligible but the first. Hilt and Hubbard (H-H) 
(44), using density matrix formalism, have developed che theory for 
nuclear magnetic relaxation of equilateral triangular configurations 

symmetry axis. They consider both the cases of a molecule undergoing 
random jumps between three equilibrium positions and a molecule under- 






le definition ol 



’.- 4 . 








experimental work on polycrystslline samples. 

Fig. 14 depicts a comparison of this theory vi 



theoretical 
multiplied by 



values better if the sum of exponentials in Eg. 
a single exponential to take into account, in an ad hoc manner th« 
effects of ineermolecular dipole-dipole interactions. One must hot 
ever assume a quite largo intermolecular proton-proton interaction 
it observed non- 





Fig. M. LN IM, RECOVERY) vs. TIME - PROTONS 









2 / EXP(-t/T°)sinSd3. 



Thla problem required the use of Simpson's numerical integration pro- 
cedure to evaluate the equivalent Integral, 

4CtJ> 8 - / EXP[-t/lJjd*, (4.22) 

where the substitution, x - cosS. uaa nnde -n,- , 

, X Msa, was mace. The integral waa approxi- 
mated by 10 Intervals, i.e., x - 0, 0.2, 0.3 ... l o and the cal 
lation was performed using a Hewlett-Packard 9100A computer. This 
integration procedure waa carried out for at least three different 
values of e for each value of w^. For a given value of « o t , the 
corresponding l‘ was available from the slope of 

■ in / EXP[-t/lJ]dx (4.12) 

vs. t. One may not rigorously define a single T? for the M (t) re- 
covery expressed by Eq. (4.11). He find however that Eq. (4.12) is a 
quite linear function of t (see Fig. 13) and . single ij governing the 

!. Log T c . The four frequencies used in the experiment 



and chat the curves corresponding to different ^ are indistinguishable 
in the short correlation time limit ((w t ) 2 « l). 

He now proceed to obtain an expression for vs. Tj. prom 

Tj is directly proportional to V but this is not a. rigorously valid 
expression. Assumption, The actual physical situation is such that, 
in the long correlation time limit, T 2 ® » kr 



i/r 




Fi} 15. LN <R( I )> s *5. TIME -i- T' 



quite well justified experimentally it 
shown by the high linearity of the pic 






L ° T '' UI “ + ioV“ ’ 

ir regression techniques Co fit t 
tnts in the linear low temperature region of Figs. 
Eq. (4.13), one obtains four values of E which s 
thin statistical error. A weighted overage over t 
is performed using as weighting factors the inver. 
a 991 confidence factor with each estimated standi 
The following value o 



sxact theory predicts 






nearly single recovery in che low 
l 4 of ref. 44), whereas in the short correlation time limit 
it true. Thus we expect our experimentally determined T^ 
accurate in the long correlation time limit. Secondly, che 
s temperature is increased, of thermally a 






grcssion fit of experimentally obtained values of LnR(t) vs. t (R(t) 




mally activated molecular 




((ms) ’j, )'aoi 




relaxation 







paper that a single exponential recovery is a very good approximation 
to the exact theory when (w 0 t.) 2 * 100. Ho observe from this figure 
that when (“ o r c ) ■ 100, * 13.2 T', where T? is the effective 

intramolecular spin-lattice relaxation time and T' is defined by 
Eg. <4. 7). For « 0 - 2x x 20.8 MBs we calculate the following values 



r - 1* - 61 






horizontal 



large separation of 






d (4.17) t, 









s sixth power of the nuclear separation. He have 
.button to range from 0.04/1^ i 

t contribute from 0.85/1®^ to 1-86/1®,,, with a most probable value 
predictions and our experimental T,** 






of 1.27/1*^.. Hi 

results, whereas the ( 0 -T) values do not, 

not due to experimental uncertainty, but 
the different physical quantity measured 



10 methods is quite possibly 
rather a manifestation of 

tioned previously. X-ray diffraction measurements give information 
concerning density peaks in the electron cloud, not the nuclear posi- 
tion as does neutron diffraction. Recall chat molecular bonding in 
NHj is such that Che molecular electron cloud is heavily concentrated 
about the nitrogen nucleus, leaving the protons relatively bare. In 
the solid each proton forms a hydrogen bond with a nearest neighbor 
molecule through mutual electrostatic attraction between Che proton 
and the lone electron pair of Che neighboring molecules. It Seems 
quite feasible that the difference synthesis, performed by Olovssen- 



Templaton, 












4.4.2. Interpretation of Proton linewidth Baca 

We presented In Sec. 3.4.1. a qualitative picture o( the phenon- 
enon of linewidth reduction in solids through raolecular notion. The 



T ■ Tj. (4.18) 

We also showed that T, is roughly equal to the inverse of the linewidth 



t„ • (6o>) -X - 



(4.19) 









transitions quite effectively. 




which the linewidth should beain to undergo a sharp reduction. The 
value obtained in this Banner (-59K) agrees quite favorably with the 
onset of the first linewidth transition as depicted in Fig. 18. We 



le linewidth transition which is c* 
id rotation of the molecules at 



le observes from Fig. 18 two additional points of Interest dis- 

in eddltional llnewidth transition, beginning 
it interrupted before completion by the solid-liquid 
•nd linewidth transition is of particular 
interest because it marfcs the activation of a motional process com- 
pletely distinct from symmetry axis hindered rotation. We speculate 

cules, or molecular aelf-diffusion-note that molecular self-diffusion 
must involve isotropic reorientation. 

made; however, one may obtain a very good estimate oi 

and Eq. (4.19). By direct analogy with 

lich Eq. (4.19) is valid, i.e., a plot of t c2 , the correlation time 
ir the second process, vs. absolute temperature will cross the (Su)~‘ 
irve of Fig. 17 at T = 180 K. This point on the graph corresponds ti 









io possible to determine 
s to an accuracy of approximately 



linewidth 



The theories 



are difficult to ui 
required. Waugh and Fc 



make suitable approximations to a very simple tl 
the following approximate relationship relating 
of a linewidth transition 







cure of the transition midpoint. This relationship agrees within 10% 
of more detailed (but not necessarily more accurate) theories. For 
comparison, Eq. (4.21) gives a value of 2.4 kcal/mole for the activa- 



Eq. (4.21) we estimate the activation energy 






The question of why tl 



provides a nice case for discussion of the relative sensitivity of the 




Ice this opportunity in answer to the question 

key to this question. The linewidth measure- 
orved to be extremely sensitive to molecular 
1 where Eq. (4.19) is approximately satisfied 



ither frequency. The llnewidth is a characteristic of the solid 
a through CWNMR techniques is fixed b] 






Eq. (4.23) is to TNMR what Eq. 



externally ■ 
nay study by TNMR is a laboratory 






only by how cleverly we design our electronic appa- 
usually accessible. A book describing these techniques in 



Me used the results of a second n 
ing the T^ data; the calculation is 
(33) lattice and molecular parameters ore used. Since 









Second neighbor protons (Protons mutually H bonded t( 




points (circled) were obtained by Gutowsky and Pake 
Clearly the most startling contrast with which 

la predicted rigid lattice value of Che second me 

Jhich is longer even than Che spontaneous transition 
•d Indicate. Clearly the Tj measurements predict no 






t, illustrated 
t as profound, is the large 









barrier. Considerable effort ha* 
many times in the literature, is 

triangular configuration of spin 



is rotating at an angular rate greater thai 
The resolution of this problem requires 

molecule through its hindering potential 
Fort has been expended at the University of 

i classical rotation of a 



e calculated rigid 






dent) the classical expression of Andrew and Bersohn should 












Assuming than all lattice distances esFsnd according 






into fij - M°/(l + 04T) 6 . This re] 



of validity, edT is so snail that terns of order higher than first 
are negligible. Thus, one has M 2 - M°/ (1 + fadt). We used the ex- 







tion of a fundamental difference in the quantum mechanical tunneling 

where it is predominantly classical) would offer some valuable infor- 

s peculate that the llnewidth transition occurs because the motion in- 
volved changes from a predominantly coherent character, at very low 



Interpretation of Nuclear Magnetic Relaxation Data 



in the number of degrees of freedom and relative magnitude of molec- 



a typical isotropic molecular reotientational time si 

rapidly to a vacent lattice position, is invalid for 
describe a strongly bonded polar liquid such as ammoi 




»**«!<■ 

^■f(4*]V 



t,- i (4f>, ■»«:«>■ 

il log of Eq. (4.25), one has an 







Rg.20. LOGJT'I vs. lOV' (TEMPERATURE) 






pessimistic + 102 uncertainty 




previously. 



(64) for Che HD free molecule (200 + 20 
conclusion is drawn thee lictle hydrogen bonding is present 



Lc hydrogen bonding in liquid annonia is completely 



work strongly supports the X- 






ly diffraction (2 












e qQ/fi only through distortion ol 













by X-ray arJ heat of vaporisation work. 






.(t) - 




Eq . (4.30) because its weighting facte 










«.32) 



Richards (68) has proved that My* ■ R(t) for the sane physical approx- 



that the relatively strong intraaolecular 



in isotropic reorientation. 



a. i . «Ai a , ..mg’* .,«(;)• ».» 






!;L 






c fora Uf-S 






We discussed qualitatively 



14 N nuclei on two molecules are not in the same state when chemical 






1 n J 2 S(S + 1) 

where S - 1, Che 14 » spin, and Uj, and w s are respeeeivcly Che Larnor 











out the multiplet structure end narrowed the observed singlet to e 
experiment Indicates that HjO is a fantastically efficient catalyst 







combine Eqs. (4.33), (4.35), (4.36), (4.38) and (4.41) Co write the 
theoretical expressions. We denote by a subscript t that these values 






(4.42) 



(4.43) 



In Sec. 4.5.1. we calculated Che rotational correlation cine vs. 

alternate approaches. We thus feel that the values calculated for I- 
using the solid state shifted value of the quadrupole coupling 




8 S 






constant (3.47 MHz) may be used for r^ in Eqs. (4.42) and (4.43) wi 

ware taken from ref. (70). Pig. 23 depicts a comparison of 
perimental results with those of Smith and Powles (43) and 

ref. (43). The actual experimental scatter of our points, taken at 
nine different temperatures, is represented as heavy vertical bars 

to minimize the experimental deviations. Mote that the theoretical 

data display. One might easily explain a slight difference in the 
absolute value of different experimental results in terms of slight 

perimental curve is approximately equal to chat of Che other curves 

has not been satisfactorily explained; possibly a clue could be ob- 
tained through an isothermal measurement of vs. sample pressure. 

Tj™. Simplification of Eq. (4.43) may be achieved by noting that 




FlgSSLOGj T, ) vs. IC{/(TEMPERATURE) 




suitable for fitting experimental data by linear regression. We use 
resenting T?P. One obtains from the linear regression fit a value 






roton chemical exchange activation energy. The error 
value is considerably lover than one might expect from 

kcal/oole. It is conceivable that the proton-proton exchange actlva- 
efflciency with which H^O impurity in destroys the proton c 



INSTRUMENTATION AND MEASUREMENT PROCEDURE 

Matheson Co. and found to be 99.9991 pure NH„. A first prerequisite 
purity. In particular, oxygen contamination was to be avoided. The 



by heating the system to above 370 K using s heat gun. This process 
pressure system. The sample system circuit remained intact throughout 

thee of ref. (43), indicating a slight impurity. The sample was 

(roughly three days), the proton T^ values increased slightly, reach- 
ing a limiting value of ~ 5 sec at 198 K. A temperature run was made 









points throughout the temperature 



samples again and taking spot 
Perfectly consistent results were obtained from both samples in the 






Fig. 



TEMPERATURE CONTROL CRYOSTAT 






surrounding the sample chamber and by providing a tic exchange ga 
with a slowly boiling cryogenic liquid, and then allowing heat ti 




as follows: 1.) Three $32 varnished copper wire leads for the heat- 
ers — electrical resistance of each is 0.6 Ohms, 2.) Four 636 varnish- 










Mueller bridge 






xwer input to the bomb must equal the energy lost 
* cryogenic bath, IE one expects an equilibrium 

f the bridge is out of balance an error voltage is in- 
Kaithley model 149. The magnitude of the error voltage 

gain setting of the Kelthley. The polarity of the error voltage is 
determined by whether the bomb is too hot or too cold relative to the 
desired temperature. If the bomb is too cool, one increases the power 

slightly different input power. We provide this slight variation in 



+10 Vdc) produces roughly an equal change In voltage across the bomb 
of the out-of-balance signal. This quadratic behavior provides ex- 
supply. A Heathklt model PS-4 regulated power supply is not a part of 
the control loop and is used as a booster to facilitate rapid changes 

sensitivity setting of the Keithley. The temperature stability fig- 
ures presented earlier are based on a sensitivity of 1 uV full scale 
with 6 V applied to the Mueller bridge. Mo loop instability was ever 






power input to the cryogenic bath and because it facilitates setting 
the average heater bias voltage. Fig. 27 illustrates tho average 

pressure in the cryostat vacuum jacket was approximately 10" 4 Torr 






V control, has a steady 







euw 




















CHAMBER 



TEMPERATURE BOMB 



value. Fig. 29 Illustrates the results of the oeasurements plotted 





















ACROSS SAMPLE (K) 







sumption, Eq. (3.69) may be used to measure T.. 
this manner after the first pulse, with M 0 ; likewise, if the 90” pulses 



v 4 C0 - v 4 (o)[i - stn-t/Tj)!. 

mental data using Bq. (5.1). One plots Ln(l - V 4 (t)/V 4 (0)] vs. t 






advantage of illustrating immediately: (1.) The precision of the 

90° pulse tuning and (2.) any non-linearity in the recovery. If the 
90° pulse is tuned correctly the straight line should pass through the 
origin, if not the intercept will be finite. One should be particu- 








■° Relaxation T: 






al field gradient; the field is largest at the center and decreases 
















The magnetic field distribution funcelon g(\») is written I 



ting echo maxima for different T , ia modulated by Che factor EXP(-t\), 
V (t) - V (0)EXF[ {-t/T.) + (5.8) 






>uld plot ln[» (t)/» (0)1 vs. t 






All values of M2 calculated were corrects! 






(5-10) 



The terms u n and H m in Eq. (5.10) are, respectively, the modulation 




peek-to-peak by the oscillator. The frequency of the rf should be 
approximately equal to that of the absorption line center frequency. 
The gating circuit, actually an Integral part of the pulsed amplifier. 



the power amplifier 










PULSE SPECTROMETER ELECTRONICS 






Oscillator 



The gated amplifier 
tor, Inc. Model PG-650-C pulsed oscillator. 

eral Radio type 805-D standard signal genera 

capable of pulses of maximum length roughly 
able droop. Twelve output coils provide a t 



. Type 805-D standard signal generator wa 

roughly +50 parts in 10° over 8 24 hour 









that this additional SI 
available through the i 



is unnecessary with the suppression now 






Hi 






Fig. 33(a). CONFIGURATION USED FOR 90’- 1 -90" PULSE SEQUENCE 



Fig. 33(b). TYPICAL DATA OUTPUT FI 



ly trigger the recording device 



6 imply sliding two switches. The meet 

from this procedure. Fig. 33(b.) illustrates the typicnl 
provided by s Tek 162. The pulse from each Tek 163 then triggers s 
Fig. 34(a.) illustrates the configuration used for a Carr-Furcell 








© . NKNNKKNKK 

I l_l l_l L.l 

© U — I — 1 — I — I I 1 I l_ 



Fig. 341b.) OUTPUT AT POINTS INDICATED IN Fig. 34 (o.) 



% 












«W\ 


a 




















:± 


i/ |VVVVua/ 





Fig. 34(e.) TYPICAL OATA OUTPUT FROM CARR- PURCELL SEQUENCE 







amplifier, b.) the wideband rf amplifier and c.) the detector cir- 

by the amplitude modulated rf carrier, and the postamplifier primarily 
The requirements placed on the detector and postampllfler are 

filtering process whereby, ideally, the detector is perfectly linear 
biased into a region of high linearity. This is done most effectively 



liodo rectification. Distortion will, of course, be intro- 
!. It oust linearly amplify the rf and have a bandwidth. 









Modifications of the ir 
amplifier and PA-620- 









tn additional 20 to 25 db gain, 
application or an rf pulse, 2.) the time t,, following Che applies- 









principle, when applied Co this situation, predicts that the uncertainty 
a - wMNn/tt ) 



g(") - i 



L - w) 



(5.12) 





.Fig. 35. TYPICAL PARALLEL TUNED SAMPLE COIL CIRCUIT 





7. ARENBURG 






illustrated in Fig. 35, but with the diodes acting like an open switch. 












of source Impedance optimization cannot be overstressed when one must 
*“ 30 usee was obtained with this circuit at 3.3 XHz; additional diode 

PA-620-L preamplifier (sec Fig. 37). The relatively strong proton re- 









This brief 






linewidth and second 



l illustrated in the operational configuration 



oscillator (99) constructed i 











FI* 38. BLOCK DIAGRAM OF THE CONTINUOUS WAVE APPARATUS 



for work with samples having long T^. 



power amplifier. The Ling model TP-100-2 






4 ‘ ikfimS 




!Ar»v- 



■ S,tKVASS.‘-’ 




Physics Teachers. 



of the candidate's supervisory corsaittee and has been approved by al 

College of Arts and Sciences and to the Graduate Council, and was ap 
proved as partial fulfillment or the requirements for the degree of 




Dean, Graduate School 

£■ H. Mad 

E. H. Hadlock